CHAPTER 1 Thermodynamics of Solids under Pressure AlbertoCopyrighted Otero de la Roza MALTA Consolider Team and National Institute for Nanotechnology, National Research Council of Canada, Edmonton, Canada Víctor Luaña MALTA Consolider Team and Departamento de Química Física y Analítica, Universidad de Oviedo, Oviedo, Spain Manuel Flórez MALTA Consolider Team and Departamento de Química Física y Analítica, Universidad de Oviedo, Oviedo, Spain Material CONTENTS 1.1 Introduction to High-Pressure Science ::::::::::::::::::::::::::::::::: 3 1.2 Thermodynamics of Solids under Pressure- ::::::::::::::::::::::::::::: 6 1.2.1 Basic Thermodynamics :::::::::::::::::::::::::::::::::::::::::Taylor 6 1.2.2 Principle of Minimum Energy :::::::::::::::::::::::::::::::::: 14 1.2.3 Hydrostatic Pressure and Thermal Pressure ::::::::::::::::::: 15 1.2.4 Phase Equilibria ::::::::::::::::::::::::::::::::::::::::::::::::: 17 1.3 Equations of State :::::::::::::::::::::::::::::::::::::::::::::::::::::: 19 1.3.1 Birch–Murnaghan Family :::::::::::::::::::::::::::::::::::::::& 20 1.3.2 Polynomial Strain Families ::::::::::::::::::::::::::::::::::::: 21 1.4 Lattice Vibrations and Thermal Models :::::::::::::::::::::::::::::::Francis 23 1.4.1 Lattice Vibrations in Harmonic Approximation :::::::::::::::: 23 1.4.2 Static Approximation ::::::::::::::::::::::::::::::::::::::::::: 26 1.4.3 Quasiharmonic Approximation ::::::::::::::::::::::::::::::::: 29 1.4.4 Approximate Thermal Models :::::::::::::::::::::::::::::::::: 32 1.4.5 Anharmonicity and Other Approaches ::::::::::::::::::::::::: 40 1.5 Conclusions ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 42 Bibliography :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 44 1.1 INTRODUCTION TO HIGH-PRESSURE SCIENCE igh pressure research [1] is a field of enormous relevance both for its sci- H entific interest and for its industrial and technological applications. Many 3 4 An Introduction to High-Pressure Science and Technology materials undergo fascinating changes in their physical and chemical charac- teristics when subjected to extreme pressure. This behavior is caused by the involvement in bonding of electrons that would otherwise not be chemically active under zero-pressure conditions (the difference between zero pressure and atmospheric pressure is negligible from the point of view of its effect on mate- rials properties, and we will use both concepts interchangeably). By and large, current chemical knowledge and the traditional rules for valence electrons are at a loss to explain most of the changes induced when materials are com- pressed, which makes chemical bonding under pressure an exciting research topicCopyrighted that has received much attention in recent years [1] (see Chapter 5). The study of pressure effects can be approached in two mostly complemen- tary ways: experimentally and computationally. In both cases, the basic object under study is the change in crystal structure a material undergoes when a given pressure and temperature are applied as shown in its thermodynamic pressure-temperature phase diagram. Experimentally, the application of tem- perature is relatively straightforward, but imposing high pressure on a sample requires specialized techniques that have been under development for the past 80 years. The field of experimentalMaterial high-pressure physics was pioneered by Percy Bridgman, who received the Nobel Prize for his efforts in 1946 [2]. There are two main experimental high-pressure techniques: dynamic com- pression methods based on shock waves, and static compression methods which make use of pressure cells. In a shock-wave compression experiment [3], a strong shock is applied to the sample and- the propagation velocities of the shock wave inside the material are measured.Taylor Very high pressures (up to 500– 1000 GPa) and temperatures (tens of thousands of K) can be attained with this technique. Static compression techniques are applied using pressure cells, notably diamond anvil cells [4]. These methods are more accurate than dy- namic techniques, but they are also limited by the pressure& scale, with the mechanical resistance of a diamond (slightly above 300 GPa)Francis being the ulti- mate upper pressure limit for the technique. High temperatures, in the range Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 of thousands of K, can be attained by laser heating, and diamond anvil cells can be coupled to other techniques: spectroscopic (infrared, Raman, x-ray) and optical. Experimental high-pressure techniques are explored in Part II of this book. From the computational point of view, in the framework of density func- tional theory (DFT), the application of pressure is relatively straightforward: one simply compresses the unit cell and calculates the applied pressure either analytically from the self-consistent one-electron states or from the volume derivative of the calculated total energy. The latter requires fitting an equation of state. Conversely, temperature effects are difficult to model because they in- volve collective atomic vibrations—called phonons—propagating throughout Thermodynamics of Solids under Pressure 5 a crystal (see Chapter 3). The simplest computational approach to model the effects of temperature is to calculate the electronic energy, assuming the nu- clear and electronic motions are decoupled (the adiabatic approximation), and then obtaining the phonon frequencies using the derivatives of said electronic energy with respect to the atomic movements. The phonon frequencies can be used in the harmonic approximation to calculate thermodynamic properties and phase stabilities at arbitrary pressure and temperature. Computational techniques for high-pressure studies are explored in this and the subsequent chapters in Part I. DespiteCopyrighted the difficulties in modeling temperature effects, theoretical ap- proaches are necessary in cases for which experiments are either not feasible or difficult to interpret. An interesting example of synergy between computa- tional and experimental approaches occurs in geophysics. Because the interior of the Earth is inaccessible, information about its composition and structure can only be inferred through indirect means [5, 6], primarily via seismic veloc- ities measured during earthquakes. Calculated data (elastic moduli, longitu- dinal and shear velocities, thermal expansion and conduction, heat capacities, Grüneisen parameters, transitionMaterial pressures, Clapeyron slopes, etc.) are very helpful in the interpretation of experimental data and in the construction of models of the Earth’s interior [7, 8]. The geophysical and astrophysical applications of high-pressure science are presented in Chapters 15 and 16, respectively. High-pressure research also has very- relevant technological applications (Chapter 14). Materials that are thermodynamicallyTaylor stable at high pressure and temperature are often metastable at zero pressure and room temperature, so it is possible to access new material phases with unique properties by the application of appropriate pressure and temperature conditions. The applica- tion of high pressure to biological samples is also of technological& relevance because it is known that the microorganism activity is diminishedFrancis or canceled by application of high pressures, a process called pascalization. In contrast to Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 the preceding applications, the pressures exerted in this case are in the order of the tenths of a GPa, much smaller than in previous examples. Pascalization can be used to increase the shelf lives of perishable foodstuffs: juice, fish, meat, dairy products, etc. This has been one of the research lines of MALTA [9–12], and is explored in Chapters 12 and 13. To summarize, high pressure is a very active field of research and one whose ramifications affect many scientific and technological fields, from astro- physics and geophysics to materials physics and the food industry. From the fundamental view, the behavior of materials at high pressure is still poorly un- derstood, and the usual textbook chemistry rules that apply to zero-pressure chemistry are essentially useless when molecules and materials are subjected 6 An Introduction to High-Pressure Science and Technology to extreme compression. This makes high-pressure research an interesting and virtually unexplored field of study. In the remainder of the present chapter, we give an overview of the basic thermodynamic principles and mathematical tools that are needed to master the rest of the book. Namely, we review the thermodynamics fundamentals (Section 1.2), the equations of state used to describe the responses of materials to compression (Section 1.3), and the har- monic approximation, the basic theoretical framework in which temperature effects are modeled computationally (Section 1.4). 1.2Copyrighted THERMODYNAMICS OF SOLIDS UNDER PRESSURE 1.2.1 Basic Thermodynamics Hydrostatic pressure, together with its extensive associated variable volume, is a fundamental quantity covered in most thermodynamics textbooks [13–17]. Pressure can be defined by considering the mechanical work exerted on a closed system (in which there is no matter exchange with the environment) through a change in its volume: Material δW = −PEdV . (1.1) In this equation, PE is the pressure applied on the system or environmental pressure [18] (also, see the subsection on irreversible pressure-volume work and references therein in Chapter 2 of Reference 15). When